I purposely left a dandelion in my plant bed last year to find out if it will die in the winter. Spring came, she came back. I dug the soil and found out that root was 12" long. I did that because I was aware that they're roots are soft, easily breaks. So mowing I guess won't kill it.

Mowing won't kill dandelions. They bloom when the hours of sunlight fall in a certain range (I used to know the range, but have forgotten). But basically, they'll bloom for a few weeks in the spring and a few weeks in the fall. The same plant can bloom multiple times in the spring and in the fall.

If you've had a few days of rain, this is the perfect time to pull the dandelions because the soil will be damp, so the root is more likely to come out than to break. If it breaks off, the dandelion will grow back.

I use a weed hound to pull them. I step down on it, give it a twist and pull straight up by the knob on top. If you pull by the handle on the side, it may loosen and not pull the whole root out.

BPGreen, It seems like the dandelions here bloom all summer long. It does seem like they all bloom at the same time in the spring.
At my old house, mowing the tops seemed to slow them down, and keep them from going to seed. The ones at the new house don't seem bothered at all by mowing.